=

Now if 8 ß, the plane contains the first line; if d = ß1, the second.

Hence, if yVaa, be the shortest vector distance between the lines, we have

S.aa, (B-B,-yVaa,) = 0,

or Ty Vaa,) =TS. (B-B1) UVaa1,

the result of § 203.

211. Find the equation of the plane, passing through the origin, which makes equal angles with three given lines. Also find the angles in question.

Let a, ẞ, y be unit-vectors in the directions of the lines, and let the equation of the plane be

Hence

dS.aßy = x(Vaẞ+ Vẞy+ Vya).
δδ.αβγ

S.p(Vaẞ+VBy+ Vya) = 0

is the required equation; and the required sine is

S.aßy

T(Vaẞ+VBy+Vya)

212. Find the locus of the middle points of a series of straight lines, each parallel to a given plane and having its extremities in two fixed lines.

the fixed lines. Also let x and x, correspond to the extremities

of one of the variable lines, being the vector of its middle point. Then, obviously,

This gives a linear relation between x and x, so that, if we substitute for a, in the preceding equation, we obtain a result of the form

where d and are known vectors. The required locus is, therefore, a straight line.

213. Three planes meet in a point, and through the line of intersection of each pair a plane is drawn perpendicular to the third; prove that, in general, these planes pass through the same line.

Let the point be taken as origin, and let the equations of the planes be

Sap

= 0, SBP = 0, Syp = 0.

The line of intersection of the first two is Vaß, and therefore the normal to the first of the new planes is

Γ.γΓαβ.

Hence the equation of this plane is

S.pV.yVaß = 0,

or Spp Say-Sap Sẞy = 0,

and those of the other two planes may be easily formed from this by cyclical permutation of a, ß, y.

We see at once that any two of these equations give the third by addition or subtraction, which is the proof of the theorem.

214. Given any number of points A, B, C, &c., whose vectors (from the origin) are a,, a,, a,, &c., find the plane through the origin for which the sum of the squares of the perpendiculars let fall upon it from these points is a maximum or minimum.

U

be the required equation, with the condition (evidently allowable)

То = 1.

The perpendiculars are (§ 208) - 'Swa,, &c.

and the condition that is a unit-vector gives

Sada = 0.

Hence, as do may have any of an infinite number of values, these equations cannot be consistent unless

The values of a are known, so that if we put

Σ.αδαπ = φω,

4 is a given self-conjugate linear and vector function, and thereforex has three values (91, 92, 93, § 164) which correspond to three mutually perpendicular values of ☎. For one of these there is a maximum, for another a minimum, for the third a maximum-minimum, in the most general case when 91, 92, 9, are all different.

215. The following beautiful problem is due to Maccullagh. Of a system of three rectangular vectors, passing through the origin, two lie on given planes, find the locus of the third.

Let the rectangular vectors be w, p, σ. ditions of the problem

Then by the con

The solution depends on the elimination of p and among these

five equations. [This would, in general, be impossible, as p

and between them involve six unknown scalars; but, as the tensors are (by the very form of the equations) not involved, the five given equations serve to eliminate the four unknown scalars which are really involved. Formally to complete the requisite number of equations we might write

Та = a, Tp = b,

but a and b may have any values whatever.]

the required equation. As will be seen in next Chapter, this is a cone of the second order whose circular sections are perpendicular to a and B. [The disappearance of x and y in the elimination instructively illustrates the note above.]

EXAMPLES TO CHAPTER VI.

1. What propositions of Euclid are proved by the mere form of the equation

p = (1-x)a+xß,

which denotes the line joining any two points in space?

2. Show that the chord of contact, of tangents to a parabola which meet at right angles, passes through a fixed point.

3. Prove the chief properties of the circle (as in Euclid, III) from the equation

where Ta =

p = a cos 0+ẞ sin 0;

TB, and Saß = 0.

4. What locus is represented by the equation

where Ta 1?

S'app2 = 0,

5. What is the condition that the lines

Γαρ = β, Γαρ = βι

intersect? If this is not satisfied, what is the shortest distance between them?

6. Find the equation of the plane which contains the two parallel lines

8. Find the equation of a straight line passing through a given point, and making a given angle with a given plane. Hence form the general equation of a right cone.